Two Paths

Time Limit: 4000/2000 MS (Java/Others)    Memory Limit: 153428/153428 K (Java/Others)
Total Submission(s): 613    Accepted Submission(s): 312

Problem Description

You are given a undirected graph with n nodes (numbered from 1 to n) and m edges. Alice and Bob are now trying to play a game. 
Both of them will take different route from 1 to n (not necessary simple).
Alice always moves first and she is so clever that take one of the shortest path from 1 to n.
Now is the Bob‘s turn. Help Bob to take possible shortest route from 1 to n.
There‘s neither multiple edges nor self-loops.
Two paths S and T are considered different if and only if there is an integer i, so that the i-th edge of S is not the same as the i-th edge of T or one of them doesn‘t exist.

Input

The first line of input contains an integer T(1 <= T <= 15), the number of test cases.
The first line of each test case contains 2 integers n, m (2 <= n, m <= 100000), number of nodes and number of edges. Each of the next m lines contains 3 integers a, b, w (1 <= a, b <= n, 1 <= w <= 1000000000), this means that there‘s an edge between node a and node b and its length is w.
It is guaranteed that there is at least one path from 1 to n.
Sum of n over all test cases is less than 250000 and sum of m over all test cases is less than 350000.

Output

For each test case print length of valid shortest path in one line.

Sample Input

2
3 3
1 2 1
2 3 4
1 3 3
2 1
1 2 1

Sample Output

5
3

Hint

For testcase 1, Alice take path 1 - 3 and its length is 3, and then Bob will take path 1 - 2 - 3 and its length is 5.
For testcase 2, Bob will take route 1 - 2 - 1 - 2 and its length is 3

才讲的K短路,第二天多校就考到这个。=-=

比较裸，直接用A*算法+Dij堆优化，不加堆优化的Dij会超时的，而且这个题数据边的w值会很大，所以所有的边的变量，中间累加的变量，数组存下的长度，最后的A*函数返回结果（一开始WA了很多次，就在这里，返回值也要long long，只顾改了变量了，白白WA了几发）都要设置成long long，第二个就是无向边了，建图和反向建图的两个数组都要push两次。这也是一开始反向建图少一个就WA。

#include <iostream>
#include <algorithm>
#include <vector>
#include <cstring>
#include <queue>
#include <cstdio>
#define MAXN 100010
#define INF (1LL<<62)
using namespace std;
typedef long long ll;
typedef pair<int, ll> P;
int N, M, S, T, K;
ll dist[MAXN];
ll tdist[MAXN];
int cnt[MAXN];
bool f[MAXN];
vector<P> Adj[MAXN];
vector<P> Rev[MAXN];
struct Edge {
    int to;
    ll len;
    Edge(){}
    Edge(int t, ll l):to(t), len(l){}
};
priority_queue<Edge> q;

bool operator<(const Edge &a, const Edge &b) {
    return (a.len + dist[a.to]) > (b.len + dist[b.to]);
}

void dijkstra() {
    memset(dist, 0, sizeof(dist));
    fill(tdist, tdist+MAXN, INF);
    tdist[T] = 0;
    while(!q.empty()) q.pop();
    q.push(Edge(T, 0));
    while (!q.empty()) {
        int x = q.top().to;
        ll d = q.top().len;
        q.pop();
        if (tdist[x] < d) continue;
        for (int i = 0; i < Rev[x].size(); i++) {
            int y = Rev[x][i].first;
            ll len = Rev[x][i].second;
            if (d+ len < tdist[y]) {
                tdist[y] = d + len;
                q.push(Edge(y, tdist[y]));
            }
        }
    }
    for (int i = 1; i <= N; i++){
        dist[i] = tdist[i];
    }
}

//注意这里是long long的返回结果啊！
ll aStar() {
    if (dist[S] == INF) return -1;
    while (!q.empty()) q.pop();
    q.push(Edge(S, 0));
    memset(cnt, 0, sizeof(cnt));
    while (!q.empty()) {
        int x = q.top().to;
        ll d = q.top().len;
        q.pop();
        cnt[x]++;
        if (cnt[T] == K) return d;
        if (cnt[x] > K) continue;
        for (int i = 0; i < Adj[x].size(); i++) {
            int y = Adj[x][i].first;
            ll len = Adj[x][i].second;
            q.push(Edge(y, d+len));
        }
    }

    return -1;
}

int main() {
//    freopen("in.txt","r",stdin);
    std::ios::sync_with_stdio(false),cin.tie(0),cout.tie(0);
    int tt;
    cin>>tt;
    while(tt--) {
    //初始化，忘了就WA
    for(int i = 0; i <= N; i++) {
        Adj[i].clear();
        Rev[i].clear();
    }
    int a, b;
    ll t;
    cin >> N >> M;
    for (int i = 0; i < M; i++) {
        cin >> a >> b >> t;
        Adj[a].push_back(make_pair(b, t));
        Adj[b].push_back(make_pair(a, t));
        Rev[b].push_back(make_pair(a, t));
        Rev[a].push_back(make_pair(b, t));
    }
    S = 1;
    T = N;
    K = 2;
//    cin >> S >> T >> K;

    if (S == T) K++;

    dijkstra();
    cout << aStar() << endl;
    }
    return 0;
}

次短路做法，用的别人模版：

原理就是在存dis最短路的时候同时用另外一个数组存次短路，在最短路交换边和dis[]值得时候，次短路存下新交换的没最短边长的值即可。

#include<cstdio>
#include<cstring>
#include<queue>
#include<functional>
#include<vector>
#include<algorithm>
using namespace std;
typedef long long int ll;
#define MAXM  350010
#define MAXN 250010
#define INF 1223372036854775807
typedef pair<ll,int> pp;

typedef struct node
{
    int v;
    ll w;
    node(int tv=0,ll tw=0):
        v(tv),w(tw){};
}node;

int n,m;
vector<node> edge[MAXM];
ll dis[MAXN],dis2[MAXN];   //dis[i]表示最短路，dis2[i]表示次短路

void solve()
{
    fill(dis+1,dis+n+1,INF);
    fill(dis2+1,dis2+n+1,INF);

    priority_queue<pp, vector<pp>, greater<pp> > q;  //用优先队列加速搜索
    dis[1]=0;
    q.push(pp(dis[1],1));   //second是该边指向（虽然是无向的）的顶点，first是这条边的权值
    while(q.size())
    {
        pp p=q.top(); q.pop();
        int v=p.second;
        ll d=p.first;
        if(dis2[v]<d) continue;  //如果当前取出的值不是到v的最短路或次短路就contniue,因为v->e的最短边和次短边一定是由->v的最短边和次短边+edge(v,e)得到
        for(int i=0;i<edge[v].size();i++)
        {
            int e=edge[v][i].v;
            ll d2=d+edge[v][i].w;
            if(dis[e]>d2)
            {
                swap(dis[e],d2);
                q.push(pp(dis[e],e));
            }
            if(dis2[e]>d2&&d2>dis[v])  //d2>dis[v]防止d2小于dis[v],这样v->e就变成负权边了，但可有可无
            {
                dis2[e]=d2;
                q.push(pp(dis2[e],e));
            }
        }
    }
    printf("%lld\n",dis2[n]);

}

int main()
{
    int t;
    scanf("%d",&t);
    while(t--)
    {
        scanf("%d%d",&n,&m);
        for(int i=1;i<=n;i++) edge[i].clear();
        for(int i=0;i<m;i++)
        {
            int a,b;
            ll c;
            scanf("%d%d%lld",&a,&b,&c);
            edge[a].push_back(node(b,c));
            edge[b].push_back(node(a,c));
        }
        solve();
    }
}